Controllability of Volterra–Fredholm type systems in Banach spaces
Introduction
In this paper we show that the examples in [1], [2] cannot be recovered as special cases of the abstract results. If the semigroup is compact, then the assumptions H2 and A2 in [1], [2], respectively, are valid if and only if X is finite dimensional. As a result, the applications are restricted to ordinary differential control systems, see also Triggiani [7] for complementary details. We remark here that there is long list of papers on exact controllability of abstract control system (including, first order systems, second order systems, abstract evolution systems and integro-differential systems) which contain a similar technical error, see for instance [1], [2], [8], [9], [10] and the references therein. Motivated by these papers we extend the results in Triggiani [7] and Henríquez [4] for a class of abstract control system which will allow us to discuss the above in detail.
Section snippets
On the lack of exact controllability in abstract control systems
In this section, we study briefly the lack of exact controllability of certain classes of abstract control systems and prove our remarks in the Introduction. To continue, we need to introduce some notations. Let , be Banach spaces. In the sequel, represent the space of bounded linear operators from W into Z endowed with the uniform operator topology, is the closed ball in Z with center at z and radius r and the closure of a set S in the space Z.
Throughout this
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